Localization for quantum groups at a root of unity

In the paper Quantum flag varieties, equivariant quantum $\mathcal{D}$-modules, and localization of Quantum groups, Backelin and Kremnizer defined categories of equivariant quantum $\mathcal{O}_q$-modules and $\mathcal{D}_q$-modules on the quantum flag variety of $G$. We proved that the Beilinson-Bernstein localization theorem holds at a generic $q$. Here we prove that a derived version of this theorem holds at the root of unity case. Namely, the global section functor gives a derived equivalence between categories of $U_q$-modules and $\mathcal{D}_q$-modules on the quantum flag variety.

For this we first prove that $\mathcal{D}_q$ is an Azumaya algebra over a dense subset of the cotangent bundle $T^\star X$ of the classical (char 0) flag variety $X$. This way we get a derived equivalence between representations of $U_q$ and certain $\mathcal{O}_{T^\star X}$-modules.

In the paper Localization for a semi-simple Lie algebra in prime characteristic, by Bezrukavnikov, Mirkovic, and Rumynin, similar results were obtained for a Lie algebra $\mathfrak{g}_p$ in char $p$. Hence, representations of $\mathfrak{g}_p$ and of $U_q$ (when $q$ is a $p$'th root of unity) are related via the cotangent bundles $T^\star X$ in char 0 and in char $p$, respectively.